PEDMAS important for what order to manipulate equations
\[
4*(y-4)+(x+1)^2=z
\]
If I give you x and y, how would you solve this equation?
Often times we want flexible equations
Prices are important in economics, but not always available for environmental goods.
How do we get prices if we know quantity?
\[
\require{cancel}
\begin{aligned}
Q&=\frac{(400-P)}{80} &\text{Isolate P in terms of Q} \\
\end{aligned}
\]
Often times we want flexible equations
Prices are important in economics, but not always available for environmental goods.
How do we get prices if we know quantity?
\[
\require{cancel}
\begin{aligned}
Q&=\frac{(400-P)}{80} &\text{Isolate P in terms of Q} \\
80Q&=\frac{(400-P)\cancel{80}}{\cancel{80}} &\text{ Multiply both sides by 80} \\
80Q-400&=\cancel{400}-\cancel{400} -P &\text{ Subtract both sides by 400} \\
-1(80Q-400)&=-P(-1) &\text{Multiply both sides by -1} \\
400-80Q&=P &\text{Flip terms for simplicity}
\end{aligned}
\]
It’s easy to make mistakes while doing algebra. Practice makes perfect
This will probably be a nasty calculation, but that is what computers are for. The order does not matter, only that the a corresponds to the square term, the b to the 1st degree term, and the c to the constant
Solutions
What happens if \(b^2-4ac\) is negative in the quadratic formula
There are no real solutions, but imaginary solutions. Can still be useful (saddle path solutions, but that’s for another class)
Rise over run can always be used to find average rate of change between two parts of a graph
Instantaneous leads us to Calculus
Team Assessment
You’re team measured the concentrations of pesticides in a lake exposed to agricultural runoff. The intern in charge of finishing the calculations ran off for the weekend leaving you all to finish their work. They left behind the following equation describing the total amount of pesticides in the lake if runoff is stopped from the farm by a new policy incentive reducing pesticides use:
\[
\begin{aligned}
&y=(8-2t)(t+2) \\
&\text{Where } y \text{ is pesticde concentration in ppb} \\
&\text{and } t \text{ is time in years}
\end{aligned}
\]
Work with your team to discuss conceptually how you would solve the following tasks.
Write out the intern’s work in a more useful equation
How long will it take for the pesticide concentration in the lake to reach zero? Since the equation is a polynomial describe why one solution is more applicable than the other.
Present your findings (Choose between a graph or table)
What is the average change in concentration from year 0 to year 4?
Explain to your client why concentrations might behave the way they were modeled.
Solution 1
Let’s FOIL out the equation so it becomes easier to graph.
Pesticides are removed from the lake at an average rate of 4 ppb per year
Solution 5
Pesticide concentrations might initially increase in the lake from residual particles in the soil being washed into the lake. Then a mixture microbial activity and other chemical process reduce the pesticides to more inert components (You will learn the actual answer in ESM 202)
